Stability analysis: Eberhart-Russell vs Perkins-Jinks

Stability analysis answers a simple question: which variety performs consistently across environments and which one swings up and down with site quality? The two textbook methods are Eberhart-Russell (1966) and Perkins-Jinks (1968). They share the same regression idea, they differ in what is regressed on what.

The shared idea

For each environment, compute an environmental index: the mean of all genotypes in that environment minus the grand mean. Plot each genotype's yield against this index. The slope (bi) and the deviation from regression (s squared d) describe stability.

Eberhart-Russell

Regress each genotype's yield on the environmental index directly.

Y_ij = mu_i + b_i * I_j + delta_ij

mu_i      = genotype i mean
b_i       = regression slope of genotype i on env index
I_j       = env index for environment j (mean - grand mean)
delta_ij  = deviation

Perkins-Jinks

Subtract the genotype mean first, then regress the deviation on the environmental index. The slope coefficient is conventionally written as beta. Numerically beta_i = b_i - 1, so the two methods give the same information but on a different scale.

(Y_ij - mu_i) = beta_i * I_j + delta_ij

beta_i = 0   means perfectly average response (b_i = 1)
beta_i > 0   above-average response to good environments (b_i > 1)
beta_i < 0   below-average response (b_i < 1)

How to read bi (E-R) or beta_i (PJ)

bi close to 1 (or beta close to 0) is an average-responding genotype, useful as a general-purpose variety. bi greater than 1 is responsive, gains more in good environments and loses more in poor ones, useful when targeting high-input systems. bi less than 1 is unresponsive, less affected by environment, useful for low-input or marginal sites.

The other half of stability: s squared d

s squared d is the residual variance after fitting the regression. A small s squared d means the regression line explains the environmental response well: predictable. A large s squared d means the genotype responds idiosyncratically: less predictable, less useful for recommendation across diverse sites.

Worked example

Six genotypes across five environments, mean yield t/ha:

gen,E1,E2,E3,E4,E5
G1,2.91,3.42,4.18,4.85,5.62
G2,3.21,3.55,4.05,4.42,4.78
G3,2.85,3.21,3.95,4.65,5.45
G4,3.42,3.61,4.12,4.55,4.81
G5,2.65,3.05,3.85,4.92,5.91
G6,3.55,3.71,4.05,4.31,4.55

Indices (env mean minus grand mean) come out negative for E1, E2, zero-ish for E3, positive for E4, E5. Fitting per-genotype regressions:

Genotype  mean    bi     s2_d   read
G1        4.20    1.13   0.012  responsive, predictable
G2        3.80    0.65   0.008  unresponsive, predictable
G3        4.02    1.04   0.018  average response, predictable
G4        4.10    0.55   0.011  unresponsive, predictable
G5        4.08    1.34   0.025  highly responsive, less predictable
G6        4.03    0.41   0.005  unresponsive, very predictable

What each method adds

Eberhart-Russell is the more common report in Indian breeding journals, with bi and s squared d quoted directly. Perkins-Jinks is more common in UK and European literature, with beta_i centred on zero. Both surface the same stability information; the choice is often driven by audience convention.

The three-quadrant rule of thumb

Plot mean yield (x-axis) against bi (y-axis). High mean and bi close to 1 is the broadly-adapted commercial release candidate. High mean and bi greater than 1 is the high-input variety. High mean and bi less than 1 is the low-input or marginal-site variety. Low mean regardless of bi is a drop. This single chart is the simplest decision aid the methods give you.

What significance tests you can run

Test bi against 1 (t-test, df = e - 2 where e is number of environments) to ask whether the slope departs from average responsiveness. Test s squared d against 0 (F-test against pooled error) to ask whether the deviation is real or just noise. Most published reports include both tests alongside the parameter estimates.

Limitations

Both methods assume the environmental index is a linear axis of environment quality. In trials with extreme stress sites this can fail: the response is non-linear, and the regression line under-fits in the tails. AMMI and GGE handle this kind of structure better. Stability methods are best treated as a complement, not a replacement, for biplot analysis.

Reporting convention

The minimum reporting set in an Indian breeding journal is: mean yield, bi, s squared d, t-test of bi against 1, F-test of s squared d against 0, and a stability category label. The Eberhart-Russell format is more common in AICRP reports; the Perkins-Jinks format is more common in international publications.